Creating, solving and using the Jacobian matrix

To know how each population growth rate changes in response to changes in the abundance each of the other population, we take the partial derivatives.

\[\mathbf J = \begin{pmatrix} \frac{\partial \dot P}{\partial P} & \frac{\partial \dot P}{\partial H} \\ \frac{\partial \dot H}{\partial P} & \frac{\partial \dot H}{\partial H} \end{pmatrix} = \begin{pmatrix} -\delta_p - \alpha_h H & -\alpha_h P \\ \alpha_h H & \alpha_h P - \delta_h \end{pmatrix}\]

We can replace the \(P\) and \(H\) in the Jacobian with the equilibria found above \(P^*, H^*\).

\[\begin{pmatrix} -\delta_p - \alpha_h \frac{1}{\delta_h}(\phi - \delta_p \frac{\delta_h}{\alpha_h}) & -\alpha_h \frac{\delta_h}{\alpha_h} \\ \alpha_h \frac{1}{\delta_h}(\phi - \delta_p \frac{\delta_h}{\alpha_h}) & \alpha_h \frac{\delta_h}{\alpha_h} - \delta_h \end{pmatrix} = \begin{pmatrix} \frac{-\alpha_h \phi}{\delta_h} & -\delta_h\\ \frac{\alpha_h \phi}{\delta_h} - \delta_p & 0 \end{pmatrix}\]

Recall the Routh-Hurwitz criterion:

\[ J_{11} + J_{22} < 0\] \(J_{11}\) is negative, so the plant population will self regulate, while \(J_{22}\) is zero, so the herbivore population does not have a density-dependence effect on itself. ???

The other part of the Routh-Hurwitz criterion is the condition:

\[ J_{11}J_{22} - J_{12}J_{21} > 0\] In the plant-herbivore context, this suggests that the plant declines due to the herbivore (\(J_{12} < 0\)) and the herbivore increases due to the plant (\(J_{21} > 0\) as long as \(\phi > \delta_p \frac{\delta_h}{\alpha_h}\)). The signs of these elements make their product negative, and help make the above condition true.

Note: the \(\phi > \frac{\delta_p \delta_h}{\alpha_h}\) term indicates: primary production \(\phi\) is high enough to support respiration/mortality losses from the plant population at the density needed to bring the herbivore production and loss into balance, or alternately, there’s only a finite herbivore standing stock if the plant standing stock in the absennce of herbivory is larger than that required to sustain steady-state herbivore standing stock.

# We can perform eigenanalysis given the parameters above.
Jac <- matrix(c(0, -a_h * phi / d_h + d_p, d_h, -a_h * phi / d_h), 
              byrow = TRUE, nr = 2)
eig <- round(eigen(Jac)[["values"]],3)

If we performed eigenanalysis on the above Jacobian matrix, we would find that the eigenvalues are complex (-0.1+0.2i, -0.1-0.2i), meaning that the populations will oscillate or cycle, with period \(\frac{2\pi}{\omega}\). * If the real parts are zero, the system will exhibit neutral stability * If the real parts are negative, the system will converge toward the equilibrium. * If the real parts are positive, the system will approach a stable oscillation away from the equilibrium.

Creating, solving and using the Jacobian matrix

To know how each population growth rate changes in response to changes in the abundance each of the other population, we take the partial derivatives.

\[\mathbf J = \begin{pmatrix} \frac{\partial \dot P}{\partial P} & \frac{\partial \dot P}{\partial H} & \frac{\partial \dot P}{\partial C} \\ \frac{\partial \dot H}{\partial P} & \frac{\partial \dot H}{\partial H} & \frac{\partial \dot H}{\partial C} \\ \frac{\partial \dot C}{\partial P} & \frac{\partial \dot C}{\partial H} & \frac{\partial \dot C}{\partial C} \end{pmatrix} = \begin{pmatrix} -\delta_p - \alpha_h H & -\alpha_h P & 0\\ \alpha_h H & \alpha_h P - \delta_h - \alpha_c C & -\alpha_c H\\ 0 & \alpha_c C & \alpha_c H - \delta_c \end{pmatrix}\]

We can replace the \(P, H, C\) in the Jacobian with the equilibria found above \(P^*, H^*, C^*\).

\[\mathbf J = \begin{pmatrix} -\delta_p - \frac{\alpha_h \delta_c}{\alpha_c} & - \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} & 0\\ \frac{\alpha_h \delta_c}{\alpha_c} & \frac{\alpha_h \phi}{\delta_p + \alpha_h \frac{\delta_c}{\alpha_c}} - \delta_h - \frac{\alpha_c }{\delta_c}(\phi - \frac{\delta_p \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \frac{\delta_h\delta_c}{\alpha_c}) & -\frac{\alpha_c \delta_c}{\alpha_c}\\ 0 & \frac{\alpha_c}{\delta_c}(\phi - \frac{\delta_p \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \frac{\delta_h \delta_c}{\alpha_c}) & \frac{\alpha_c \delta_c}{\alpha_c} - \delta_c \end{pmatrix}\] \[\Longrightarrow \mathbf J = \begin{pmatrix} -\delta_p - \frac{\alpha_h\delta_c}{\alpha_c} & - \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} & 0\\ \frac{\alpha_h \delta_c}{\alpha_c} & 0 & -\delta_c\\ 0 & \frac{\alpha_h \phi}{\delta_p + \frac{\alpha_h \delta_c}{\alpha_c}} - \delta_h & 0 \end{pmatrix}\]

Recall the Routh-Hurwitz criterion, assuming we can update simply for three species?:

\[ J_{11} + J_{22} + J_{33}< 0\] \(J_{11}\) is negative, so the plant population will self regulate, while \(J_{22}\) and \(J_{33}\) are zero, so the herbivore and carnivore populations do not have a density-dependence effect on themselves. ???

The other part of the Routh-Hurwitz criterion is the condition (how to update for three species?):

\[ J_{11}J_{22} - J_{12}J_{21} > 0\]

# We can perform eigenanalysis given the parameters above.
Jac3 <- matrix(
  c(-d_p - a_h*d_c / a_c,            - (a_h*phi) * (d_p + a_h*d_c / a_c),      0,
           a_h*d_c / a_c,                                              0,   -d_c,
                       0,    (a_h * phi) * (d_p + (a_h*d_c) / a_c) - d_h,     0), 
              byrow = TRUE, nr = 3)
eig3 <- round(eigen(Jac3)[["values"]],3)

When we performed eigenanalysis on the above Jacobian matrix, we find that the eigenvalues are complex (0.644+0i, -0.577+0.219i, -0.577-0.219i), meaning that the populations will oscillate or cycle. What does it mean that one eigenvalue is not complex? * If the real parts are zero, the system will exhibit neutral stability * If the real parts are negative, the system will converge toward the equilibrium. * If the real parts are positive, the system will approach a stable oscillation away from the equilibrium.